Simplifying Complex Fractions: A Step-by-Step Guide
Complex fractions might seem daunting at first glance, but with a systematic approach, they can be simplified just like any other fraction. This guide will walk you through the process of simplifying the complex fraction:
(3/(x-1) - 4) / (2 - 2/(x-1))
We'll break down each step to make it easy to understand and apply to similar problems. So, let’s dive in and conquer this mathematical challenge!
Understanding Complex Fractions
Before we jump into solving, let's define what a complex fraction actually is. In essence, a complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. Think of it as a fraction within a fraction. These types of expressions often appear more complicated than they actually are, and with the right techniques, they can be simplified into a more manageable form.
Our goal here is to transform the given complex fraction into a simple fraction, where both the numerator and the denominator are single polynomials. This involves a series of algebraic manipulations, primarily focusing on finding common denominators and combining like terms. The beauty of mathematics lies in its structured approach to problem-solving, and simplifying complex fractions is a perfect example of this. By following a step-by-step method, you can unravel even the most intricate expressions. Remember, the key to mastering any mathematical concept is practice. So, as we walk through this example, try to apply the same principles to other similar problems. The more you practice, the more comfortable and confident you'll become in handling complex fractions.
Step 1: Simplify the Numerator
The numerator of our complex fraction is (3/(x-1) - 4). To simplify this, we need to combine these two terms. The first term is a fraction, and the second term is a whole number. To combine them, we need to express the whole number as a fraction with the same denominator as the first term. In this case, the denominator we're aiming for is (x-1).
So, we rewrite 4 as 4(x-1)/(x-1). Now, both terms in the numerator have the same denominator, allowing us to combine them. The expression becomes:
3/(x-1) - 4(x-1)/(x-1)
Now we can combine the numerators over the common denominator:
[3 - 4(x-1)] / (x-1)
Next, distribute the -4 across the (x-1) term:
[3 - 4x + 4] / (x-1)
Finally, combine the constant terms in the numerator:
(-4x + 7) / (x-1)
So, the simplified form of the numerator is (-4x + 7) / (x-1). This step is crucial because it transforms the numerator from a sum of terms into a single fraction, which is a key part of simplifying the overall complex fraction. Remember, the goal is to consolidate the expression into a form where we can easily perform division, which is essentially what simplifying a complex fraction involves. By focusing on one part at a time – in this case, the numerator – we break down the larger problem into smaller, more manageable steps. This approach is not only effective for complex fractions but also for various other mathematical problems. Keep practicing these steps, and you'll find yourself simplifying these expressions with ease!
Step 2: Simplify the Denominator
Now, let's shift our focus to the denominator of the complex fraction, which is (2 - 2/(x-1)). Similar to what we did with the numerator, our goal here is to combine these terms into a single fraction. We have a whole number, 2, and a fraction, 2/(x-1). To combine them, we need to express the whole number as a fraction with the same denominator as the second term, which is (x-1).
We rewrite 2 as 2(x-1)/(x-1). Now, the expression becomes:
2(x-1)/(x-1) - 2/(x-1)
Since both terms now have the same denominator, we can combine the numerators:
[2(x-1) - 2] / (x-1)
Distribute the 2 across the (x-1) term:
(2x - 2 - 2) / (x-1)
Combine the constant terms in the numerator:
(2x - 4) / (x-1)
We can further simplify this by factoring out a 2 from the numerator:
2(x - 2) / (x-1)
So, the simplified form of the denominator is 2(x - 2) / (x-1). This step mirrors the process we used for the numerator, emphasizing the importance of finding a common denominator to combine terms. By simplifying both the numerator and the denominator separately, we make the next step – dividing the two fractions – much more straightforward. This methodical approach is a hallmark of effective problem-solving in mathematics. By breaking down a complex fraction into its components and simplifying each one individually, we pave the way for a clearer and more manageable solution. Keep practicing these techniques, and you'll find that even the most challenging expressions can be tamed!
Step 3: Divide the Simplified Numerator by the Simplified Denominator
Now that we've simplified both the numerator and the denominator, we can proceed with dividing the two fractions. Recall that dividing by a fraction is the same as multiplying by its reciprocal. Our simplified numerator is (-4x + 7) / (x-1), and our simplified denominator is 2(x - 2) / (x-1). So, we'll multiply the numerator by the reciprocal of the denominator.
The expression becomes:
[(-4x + 7) / (x-1)] * [(x-1) / 2(x - 2)]
Notice that we're multiplying two fractions together. When multiplying fractions, we multiply the numerators together and the denominators together:
[(-4x + 7) * (x-1)] / [(x-1) * 2(x - 2)]
Now, we look for opportunities to simplify by canceling out common factors. We can see that (x-1) appears in both the numerator and the denominator, so we can cancel them out:
(-4x + 7) / 2(x - 2)
At this point, we've successfully simplified the complex fraction. There are no more common factors to cancel, and we've expressed the original complex fraction as a single, simplified fraction. This step highlights the power of reciprocals in division and the importance of identifying and canceling common factors to simplify expressions. The ability to manipulate fractions and recognize opportunities for simplification is a fundamental skill in algebra and beyond. By mastering these techniques, you'll be well-equipped to tackle a wide range of mathematical problems. Keep practicing, and you'll find that these manipulations become second nature!
Step 4: Final Simplified Expression
After performing the division and canceling out common factors, we arrive at the final simplified expression:
(-4x + 7) / 2(x - 2)
This matches option B in the given choices. Therefore, the expression equivalent to the original complex fraction is (-4x + 7) / 2(x - 2).
This final step underscores the importance of careful and methodical simplification. By breaking down the problem into smaller, manageable steps, we were able to navigate the complexities of the complex fraction and arrive at a clear and concise solution. The journey from the initial complicated expression to this simplified form demonstrates the power of algebraic manipulation and the beauty of mathematical simplification. Remember, the key to success in mathematics is not just finding the answer, but also understanding the process. By grasping the underlying principles and techniques, you can confidently tackle similar problems and deepen your understanding of mathematical concepts. So, keep practicing, keep exploring, and keep simplifying!
Conclusion
Simplifying complex fractions involves a series of steps, including finding common denominators, combining terms, and dividing fractions. By following these steps carefully, you can transform a seemingly complicated expression into a more manageable form. Remember to practice these techniques to build your confidence and skills in algebraic manipulation.
For further exploration of fraction manipulation and algebraic simplification, visit a trusted resource like Khan Academy's Algebra I section.
